Recent questions tagged isi2024-mcs-pca

The roots of the equation $x^{4}+x^{3}+x^{2}+x+1= 0$ arethe vertices of a square.(some of) the vertices of a regular pentagon.(some of) the vertices of a regular hexagon....

Let $C=\left\{\binom{1}{2},\binom{2}{1}\right\}$ be a basis of $\mathbb{R}^{2}$ and $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be defined by $T\binom{x}{y}=\binom{x+y...

Which of the following pairs of regular expressions are not equivalent?$x x^{\star}$ and $x^{\star} x$$x(y x)^{\star}$ and $(x y)^{\star} x$$x(x x)^{\star}$ and $(x x)^{\...

Consider the disc on the $(x, y)$ plane with centre at $(0,1)$ and radius 1 unit. Let $A$ be the highest point on the disk, that is, $(0,2)$. If this disk rolls on the $x...

Let $f:[0,1] \rightarrow(0, \infty)$ be a continuous function. Define $g(x)=\int_{0}^{x} f(t) d t$ for $0 \leq x \leq 1$. If $g(1)=1$, the equation $g(x)=1 / 2$may not ha...

Let $A=\left\langle a_{1} a_{0}\right\rangle$ and $B=\left\langle b_{1} b_{0}\right\rangle$ be two 2-bit numbers and $C=\left\langle c_{3} c_{2} c_{1} c_{0}\right\rangle$...

Let $A=\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ be a set of $n$ natural numbers and let $n$ be odd. For any finite set $X \subseteq \mathbb{N}, \operatorname{Max}(X)$ ...

The $\lim _{n \rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$$(\mathrm{A})$ is equal to $0$ .is equal to $2$is equal to $1$.does not exist.

Suppose $A$ is an $n \times n$ matrix for some $n \geq 2$. Then $A^{n}=0$ if and only if and only if.$A=0$all the eigenvalues of $A$ are zero.$A$ is singular.$\operatorna...

Let $A_{1}$ and $A_{2}$ be two arbitrarily chosen subsets of $\{1,2, \ldots, n\}$. The number of ways such that $A_{1} \cap A_{2}=\emptyset$ is$2^{n}$$4^{n}-3^{n}$$3^{n}$...

The following program Check takes as input a graph and an integer and outputs True or False.function $\operatorname{CHECK}(G=(V, E), k)$if for all vertices $u$ and $v$ in...

The maximum value of the function $f(x)=\frac{x}{2}-\frac{1}{1+x^{2}}$ on the set $\{x \in \mathbb{R}: x(x-1) \leq 12\}$ is$0$$33 / 17$$-8 / 5$$-1$ 

If $P$ and $Q$ are two propositions, then the expression $(P \wedge(P \rightarrow Q)) \rightarrow Q$ isalways true.true for only one assignment.always false.false for onl...

If\[2021^{2024}=100 m+n\]for some $m \in\{0,1,2, \ldots\}$ and $n \in\{0,1, \ldots, 99\}$, then the value of $n$ is$21$$31$$61$$81$ 

Let $G$ be a simple undirected graph. For a vertex $u \in V(G)$, let $N_{G}(u)$ denote the set of neighbours of $u$ in $G$, i.e., $N_{G}(u)=\{v: u v \cdot \in E(G)\}$. Le...

What is the maximum number of nonzero values in the adjacency matrix of a directed graph that has $36$ vertices, but no self-loops?$36$$180$$630$$1260$ 

Let $A$ and $B$ be two real symmetric square matrices with the property that both $A$ and $B$ have the same set of eigenvectors. Then, which of the following is always tr...

Consider the following pseudo-code.function $f(n, m)$if $m=0$ thenreturn 1if $m=1$ thenreturn $n$Return $f\left(n,\left\lfloor\frac{m}{2}\right\rfloor\right) \cdot f\left...

Consider the curves$$\begin{aligned}& y_1(x)=\frac{1}{2} x+\frac{1}{2} x^5 \\& y_2(x)=\frac{1}{2} x^3+\frac{1}{2} x^7 \\& y_3(x)=1+\frac{1}{2} x^3+\frac{1}{2} x^7\end{ali...

Which of the following regular expressions is equivalent to the non-deterministic finite automaton shown in the figure below? $\left(a^{*} b^{*} a\right)^{*}\left(\epsilo...

There are two coins in a box. One of the coins is a fair coin and for the other, the probability of head is $\frac{2}{3}$. A coin is chosen at random from the box and tos...

Let $S=\left\{\left(\begin{array}{cc}a & b \sqrt{5} \\ -b \sqrt{5} & a\end{array}\right): a, b \in \mathbb{Z}\right\}$. The number of $A \in S$ such that $A^{-1}$ is also...